相対論的流体力学方程式の微視的導出に関する最近の試み

13th Heavy-Ion Café　「相対論的流体力学と高エネルギー重イオン反応：来し方行く

末」　 2009 年１１月１４日 , 東大理

国広悌二　（京大理）

--- 津村 - 大西氏との共同研究を中心に　 ----

Introduction• Relativistic hydrodynamics for a perfect fluid is widely and successfully

used in the RHIC phenomenology. T. Hirano, D.Teaney, ….• A growing interest in dissipative hydrodynamics. hadron corona (rarefied states); Hirano et al …

Generically, an analysis using dissipative hydrodynamics is needed even to show the dissipative effects are small.

A.Muronga and D. Rischke; A. K. Chaudhuri and U. Heinz,; R. Baier, P. Romatschke and U. A. Wiedemann; R. Baier and P. Romatschke (2007)and the references cited in the last paper.

is the theory of relativistic hydrodynamics for a viscous fluidfully established?

However,

The answer isNo!

unfortunately.

Fundamental problems with relativistic hydro-dynamical equations for viscous fluid

a. Ambiguities in the form of the equation, even in the same frame and equally derived from Boltzmann equation: Landau frame; unique, Eckart frame; Eckart eq. v.s. Grad-Marle-Stewart eq.; Muronga v.s. R. Baier et al

b. Instability of the equilibrium state in the eq.’s in the Eckart frame, which affects even the solutions of the causal equations, say, by Israel-Stewart. W. A. Hiscock and L. Lindblom (’85, ’87); R. Baier et al (’06, ’07)

c. Usual 1st-order equations are acausal as the diffusion eq. is, except for Israel-Stewart and those based on the extended thermodynamics with relaxation times, but the form of causal equations is still controversial.

---- The purpose of the present talk ---For analyzing the problems a and b first,we derive hydrodynaical equations for a viscous fluid from Boltzmann equationon the basis of a mechanical reduction theory (so called the RG method) and a natural ansatz on the origin of dissipation.We also show that the new equation in the Eckart frame is stable.We emphasize that the definition of the flow and the physical nature of therespective local rest frame is not trivial as is taken for granted in the literature,which is also true even in the second-order equations.

Typical hydrodynamic equations for a viscous fluid

Fluid dynamics = a system of balance equations

Eckart eq.Eckart eq.

energy-momentum ：energy-momentum ：

number ：number ：

Landau-Lifshits Landau-Lifshits

no dissipation in the number flow;

no dissipation in energy flow

Describing the flow of matter.

describing the energy flow.

with transport coefficients:

Dissipative part

with

--- Involving time-like derivative ---.

--- Involving only space-like derivatives ---

; Bulk viscocity,

;Heat conductivity

; Shear viscocity

--- Choice of the frame and ambiguities in the form ---

0,T u

0u N

No dissipativeenergy-densitynor energy-flow.No dissipativeparticle density

Compatibility of the definitions of the flow and the nature of the Local Rest Frame

In the kinetic approach, one needs a matching condition.

Seemingly plausible ansatz are;

Distribution function in LRF:

Non-local distribution function;

Is this always correct, irrespective of the frames?In particular,is particle frame the same local equilibrium state as the energy frame?

c.f. D. Rischke nucl-th/9809044

J. M. Stewart, ``Non-Equilibrium Relativistic Kinetic Theory”,

Lecture Notes in Physics 10 (Springer-Verlag), 1971

c.f. D. Rischke nucl-th/9809044

These issues have not been seriously considered and are obscure in the existing literature.

C. Marle, A.I.H.Poincare, 10 (1969)

The separation of scales in the relativistic heavy-ion collisions

Liouville Boltzmann Fluid dyn.

Hamiltonian

　

Slower dynamics

on the basis of the RG method; Chen-Goldenfeld-Oono(’95),T.K.(’95)

C.f. Y. Hatta and T.K. (’02) , K.Tsumura and TK (’05); Tsumura, Ohnishi, T.K. (’07)

Navier-Stokes eq.Navier-Stokes eq.

（力学系の縮約）

Relativistic Boltzmann equation

Conservation law of the particle number and the energy-momentum

H-theorem.

The collision invariants, the system is local equilibrium

Maxwell distribution (N.R.)Juettner distribution (Rel.)

perturbationperturbation

Ansatz of the origin of the dissipation= the spatial inhomogeneity, leading to Navier-Stokes in the non-rel. case . would become a macro flow-velocity

Derivation of the relativistic hydrodynamic equation from the rel. Boltzmann eq. --- an RG-reduction of the dynamicsK. Tsumura, T.K. K. Ohnishi; Phys. Lett. B646 (2007) 134-140

c.f. Non-rel. Y.Hatta and T.K., Ann. Phys. 298 (’02), 24; T.K. and K. Tsumura, J.Phys. A:39 (2006), 8089

time-like derivative space-like derivative

Rewrite the Boltzmann equation as,

Only spatial inhomogeneity leads to dissipation.

Coarse graining of space-time

RG gives a resummed distribution function, from which and are obtained.

Chen-Goldenfeld-Oono(’95),T.K.(’95), S.-I. Ei, K. Fujii and T.K. (2000)

Solution by the perturbation theory0th0th

0th invariant manifold

“slow”“slow”

Five conserved quantities

m = 5m = 5

Local equilibrium

reduced degrees of freedom

written in terms of the hydrodynamic variables.Asymptotically, the solution can be written solelyin terms of the hydrodynamic variables.

1st1st

Evolution op. ：inhomogeneous ：

The lin. op. has good properties:

Collision operatorCollision operator

1.1. Self-adjointSelf-adjoint

2.2. Semi-negative definiteSemi-negative definite

3.3.

has 5 zero modes 、 other eigenvalues are negative.

Def. inner product:

metricmetric

fast motionto be avoidedfast motion

to be avoided The initial value yet not

determined The initial value yet not

determined

Modification of the manifold ：

Def. Projection operators:

eliminated by the choice

fast motionfast motionThe initial value not yet determinedThe initial value not yet determined

Second order solutions

with

Modification of the invariant manifold in the 2nd order;

eliminated by the choice

Application of RG/E equation to derive slow dynamics

Collecting all the terms, we have;

Invariant manifold (hydro dynamical coordinates) as the initial value:

The perturbative solution with secular terms:

found to be the coarse graining condition

Choice of the flow

RG/E equation

The meaning of

The novel feature in the relativistic case;; eg.

References on the RG/E method:• T.K. Prog. Theor. Phys. 94 (’95), 503; 95(’97), 179• T.K.,Jpn. J. Ind. Appl. Math. 14 (’97), 51• T.K.,Phys. Rev. D57 (’98),R2035• T.K. and J. Matsukidaira, Phys. Rev. E57 (’98), 4817• S.-I. Ei, K. Fujii and T.K., Ann. Phys. 280 (2000), 236• Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24• T.K. and K. Tsumura, J. Phys. A: Math. Gen. 39 (2006),

8089 (hep-th/0512108)• K. Tsumura, K. Ohnishi and T.K., Phys. Lett. B646 (2007),

134

L.Y.Chen, N. Goldenfeld and Y.Oono, PRL.72(’95),376; Phys. Rev. E54 (’96),376.

C.f.

The distribution function;

produce the dissipative terms!

Notice that the distribution function as the solution is representedsolely by the hydrodynamic quantities!

A generic form of the flow vector

： a parameter ： a parameter

1 2

2 3P

P P P

Projection op. onto space-like traceless second-rank tensor;

Landau frameand Landau eq.!Landau frame

and Landau eq.!

Examples

T

satisfies the Landau constraints

0, 0u u T u T

0u N

Bulk viscosity

Heat conductivity

Shear viscosity

C.f. Bulk viscosity may play a role in determining the acceleration of the expansion of the universe, and hence the dark energy!

-independentpc.f.

( )p pa In a Kubo-type form;

with the microscopic expressions for the transport coefficients;

Eckart (particle-flow) frame:

Setting

=

= with

(ii) Notice that only the space-like derivative is incorporated.(iii) This form is different from Eckart’s and Grad-Marle-Stewart’s, both of which involve the time-like derivative.

c.f. Grad-Marle-Stewart equation;

(i) This satisfies the GMS constraints but not the Eckart’s.

i.e.,

Grad-Marle-Stewart constraints

Landau equation:

3u T u X

Which equation is better, Stewart et al’s or ours?

The linear stability analysis around the thermal equilibrium state.

c.f. Ladau equation is stable. (Hiscock and Lindblom (’85))

K.Tsumura and T.K. ;Phys. Lett. B 668, 425 (2008).

The stability of the equations in the “Eckart(particle)” frame

(i) The Eckart and Grad-Marle-Stewart equations gives an instability, which has been known, and is now found to be attributed to the fluctuation-induced dissipation, proportional to .(ii) Our equation (TKO equation) seems to be stable, being dependent on the values of the transport coefficients and the EOS.

K. Tsumura and T.K. ,PLB 668, 425 (2008).

The stability of the solutions in the “Eckart (particle)” frame:

Du

The numerical analysis shows that, the solution to our equation is stable at least for rarefied gasses.

See also,Y. Minami and T.K., Prog. Theor. Phys. 122, No.4 (2009); arXiv:0904.2270 [hep-th]

Compatibility with the underlying kinetic equations?

Eckart constraints are not compatible with the Boltzmann equation, as proved

in K.Tsumura, T.K. and K.Ohnishi;PLB646 (’06), 134.

Collision operatorCollision operator

has 5 zero modes:

Proof that the Eckart equation constraints can not be compatible with the Boltzmann eq.

Preliminaries:

The dissipative part; =

with

where

due to the Q operator.

The orthogonality condition due to the projection operator exactly corresponds to the constraints for the dissipative part of the energy-momentum tensor and the particle current!

i.e., Landau frame,

i.e., the Eckart frame,

4,

(C)

Constraints 2, 3Constraint 1

Contradiction!

Matching condition!

Phenomenological Derivation

particle frame

energy frame

Generic form of energy-momentum tensor and flow velocity:

with

Notice;

natural choice and parametrization

3e X p X cf.

K. Tsumura and T.K., arXiv:0906.0079[hep-ph]

From

In particle frame;

With the choice,

we have f_e, f_n can be finite,not in contradiction withthe fundamental laws!

Energy frame:

coincide with the Landau equation with f_e=f_n=0.

Microscopic derivation gives the explicit form of f_e and f_n in each frame:

particle frame;

energy frame;

Israel-Stewart equations fromKinetic equation on the basis of

the RG method

K. Tsumura and T.K., arXiv:0906.0079[hep-ph]

Geometrical image of reductionof dynamics

nR

t X

M

dim M m n

dim X n

( )ts

O dim ms

Invariant and attractive manifold

( )d

dt

XF X

( )d

dtsG s

M={ ( )}X X X s

( , )fX r p ; distribution function in the phase space (infinite dimensions)

{ , , }u T ns ; the hydrodinamic quantities (5 dimensions), conserved quantities.

eg.

The viscocities are frame-independent, in accordance with Lin. Res. Theory.

However, the relaxation times and legths are frame-dependent.

The form is totally different from the previous ones like I-S’s,And contains many additional terms.

contains a zero mode of the linearizedcollision operator. 2p p m

Conformal non-inv.gives the ambiguity.

0

0u u

K. Tsumura and T.K., arXiv:0906.0079[hep-ph]

For the details, see

Summary

• The (dynamical) RG method is applied to derive generic 1st- and 2nd-order fluid dynamic equations, giving new constraints in the particle frame, consistent with a general phenomenological derivation.

• The new equation in the particle frame does not show a pathological behavior as Eckart eq. does. This means that the acausality problem and instability problem are due to different origins, respectively.

Backups

/ /vC T t q x

Fourier’s law; /q T x

Then 2/vC T t T

Causality is broken; the signal propagate with an infinitespeed.

Modification;

Nonlocalthermodynamics

Memory effects; i.e., non-Markovian

Derivation(Israel-Stewart) ： Grad’s 14-moments method

+ ansats so that Landau/Eckart eq.’s are derived.

Problematic

The problem of acausality:

Extended thermodynamics

Five integral const’s ；

zero mode

pseudo zero mode sol.

Init. value

Constraints;Orthogonality condition with the zero modes

zero mode pseudo zero mode

Eq. governing the pseudo zero mode;

collision invariants

eqqpq

eqppq fAfL 1Lin. Operator;

and

Derivation of the secnd-order equationsK. Tsumura and T.K., arXiv:0906.0079[hep-ph]

with the initial cond.;

Def.

Projection to the pseudo zero modes ；

Thus,

Up to 1st order;

Initial condition ；（ Invariant manifold ）

RG/E equation

Slow dynamics (Hydro dynamics ）

Include relaxation equations

Explicitly ；

Specifically,

Def.

New!

For the ve ｌ ocity field,

0 ; Landau, / 2 ;Eckart

Integrals given in termsof the distribution function

Example: Energy frame

Frame dependence of the relaxation times

Calculated for relativistic ideal gas with

; frame independent